Distinction between graphene states and surface state contributions
In order to access the electronic properties of graphene, atomically resolved dI/dV maps were acquired close to the area used for the Au surface state mappings [small dashed rectangle in Figure 7.7 (a)]. The atomically resolved topographic STM image (b) of the mapping area clearly shows the Shockley partial dislocation lines of the Au(111) surface, the moiré superstructure and the honeycomb lattice. The STM topography reflects the perfectly intact structure of the GNFs. Defects and grain boundaries in graphene on Au lead to intense scattering and are therefore imaged even on large scale STM topographs.
Only two single defects are present in the mapping area and underline the high structural quality.
The corresponding atomically resolved dI/dV mapping in Figure 7.8 (a) and the cor-responding FFT (b) show several features on different length scales. Feature A resem-bles a circle around the center of the FFT with diameter of∼6 nm−1 and corresponds to the QPIs of the Au surface state analyzed before. Comparable qualitative appear-ance of the real space Au surface state standing waves can be recognized already in the dI/dV mapping and becomes even clearer after an inverse Fourier transform selec-tively considering only the ring-like contour of feature A [Figure 7.8 (c)]. As discussed in the previous section, these standing waves are a feature of the surface state of the underlying Au(111) substrate, which decays into vacuum exponentially [165] but does not contain any graphene related information. More importantly, the atomically re-solved dI/dV mappings show additional features labeled B, C and D connected to the
(a) Topography
G/Au Au
Au G/Au
b
50 nm
Topography (b)
10 nm
Figure 7.7 | Topographic overview of the GNF used for mappings. (a)STM topography of an ap-proximately 400×160 nm2largeR0◦GNF. Dashed rectangles indicate the areas where the Au(111) surface state mapping and the atomically resolved mappings were performed.(b)Topography of the area used for dI/dV mappings. Graphene moiré, herringbone reconstruction and atomic lattice are visible. Scan parameters: 54×54 nm2,V=-20 mV,I=1 nA,Vmod=2 mV,T=7.7 K.
7.2 Dispersion relations of single and double layer graphene 115
Figure 7.8|Quasiparticle interferences in atomically resolved mappings on a large R0◦ GNF. (a) dI/dV map at a bias volt-age of -20 mV.(b)Fast Fourier transform of the dI/dV map. The dashed hexagons in-dicate the reciprocal lattice (large hexagon) and the (p3×p3)R30◦superstructure (Bril-louin zone) of graphene (small hexagon), re-spectively. (c)Selective inverse FFTs of the features indicated in the FFT. A: Au(111) sur-face state, B: Intravalley scattering, C: In-tervalley scattering, D: honeycomb lattice of graphene without moiré and herringbone su-perstructure spots. Scanning parameters:
graphene atomic resolution. The outermost hexagon (D) connects the reciprocal lat-tice points of graphene and the selective inverse FFT shows the corresponding graphene honeycomb lattice, where the atoms appear slightly elongated due to the specific tip characteristics. The reciprocal lattice spots are surrounded by a number of sharp spots which arise due to the superposition of G/Au moiré and herringbone reconstruction [compare section 6.2.1]. The smaller hexagon (C) connects six ring-like features rotated by 30◦compared to the first order atomic lattice spots. This rotated hexagon gives rise to the (p
3×p
3)R30° superstructure in the real space images and was previously con-nected to intervalley scattering in literature [74, 77, 160, 72]. In the inverse FFT, the ring-like features C give rise to long wavelength intensity oscillations imprinted on the (p
3×p
3)R30° lattice. Finally, the small ring-like feature B in the middle of the FFT gives rise to long wavelength oscillations similar to the long wavelength oscillations modu-lating the (p
3×p
3)R30° superstructure in feature C. The latter feature is related to the intravalley scattering in graphene [74, 77, 160, 72].
k y (1/nm)
Figure 7.9|Possible scattering processes in graphene.(a)Schematic illustration of the graphene dispersion relation. (b)The combined CEC of graphene and the Au(111) surface state is depicted.
Possible transitions between graphene states are intervalley (grey) and intravalley (orange) scattering processes. For the spin-orbit split Au surface state only transitions from one to the other contour are observed (blue).(c)The scattering leads to ring-like features in the FFT of dI/dV maps with interval-ley scattering located at positions corresponding to the corners of the (p3×p
3)R30◦superstructure.
Intravalley scattering on the other hand is located at the center of the FFT (small ring). Au(111) sur-face state backscattering is observed simultaneously in the middle of the FFT (larger ring). Scanning parameters: 54×54 nm2,V=-20 mV,I=1.0 nA,Vmod=2 mV,T=7.7 K.
The dispersion relation Focusing on thep
3×p
3R30◦reflexes allows the quantitative evaluation of the G/Au dis-persion relation. The atomically resolved dI/dVmappings were acquired with real space dimensions of 54×54 nm2 in order to give sufficiently largek-resolution with kmin = 2π/L=0.1 nm−1and Lbeing the width of the real space mapping area. The large k-resolution is sufficient to resolve the structure within the intervalley (C) and intravalley (B) features. A magnification of the discussed FFT is depicted in Figure 7.9 (c) and shows a ring-like intensity distribution within the features B and C. Both features are related to QPIs of graphene electrons [74, 160, 72] as described in the following. QPIs arise due to the elastic impurity scattering of electrons between states~kand states~k0on the constant energy contour (CEC) [Figure 7.9 (a-b)]. In the case of the scattering between two neigh-boring K and K’ points (intervalley), the resulting scattering vector points at the corner of the hexagonal (p
3×p
3)R30◦superstructure in the FFT in Figure 7.9 (c), i.e. the corners of the Brillouin zone. As discussed in section 3.3.3 according to stationary phase
approx-7.2 Dispersion relations of single and double layer graphene 117
Figure 7.10 | Dispersion relation of G/Au(111) from intervalley scattering. (a)Magnifications of the top left intervalley scattering disc for various energies (3.5×3.5 nm-2). (b)Radial distribution of intensity after an azimuthal integration around the center of the rings in the FFTs.(c)The G/Au(111) surface state dispersion from the atomically resolved mappings (blue triangles) is plotted in combina-tion with the full surface state dispersion for pristine Au(111) (cyan diamonds) as well as for G/Au(111) (blue triangles) from Figure 7.5. The dispersion relation of the graphene electrons is reconstructed from intervalley scattering features (red squares) determined from a series of atomically resolved constant-energy mappings. The corresponding fit (red line) including uncertainty (red dotted lines).
The plottedk values are measured with respect to theΓ-point in the case of the surface state and with respect to the K-point in case of graphene. Reprinted with permission from [218]. Copyright 2014 American Chemical Society.→Online.
imation, constructive interference of the LDOS is only reached for antiparallel gradients
∇~kE(~k) corresponding to opposite group velocitiesv= ħ−1∇~kE(~k) of the involved Bloch waves [175, 174, 177]. In the case of graphene, this corresponds to scattering between two points on opposite sides of the circular CEC at adjacent corners of the Brillouin zone.
The scattering vector~q=~k0−~k for graphene intervalley scattering processes between graphene states at K02and K3amounts to
~qinter=(# –
ΓK02+~kG0 )−(# –
ΓK3+~kG) (7.1)
=# –
ΓK1−2~kG, (7.2)
where~kG= −~kG0 was introduced to fulfill the antiparallel gradient condition. Considering all possible scattering vectors connecting allowed transitions between the neighboring K
and K’ points on the CEC, six circular contours in the FFT are formed with radius
¯
¯
¯~qinter−# – ΓK¯
¯
¯=2kG (7.3)
of the ring-like feature. Note that the valuekG is measured with respect to the corre-sponding Brillouin zone corner. Similar, the transition between states on opposite sides of one constant energy circle (intravalley) will also give rise to a scattering vector, in this case around the center of the Fourier transform, with radius¯
¯~qintra
¯
¯=2kG in analogy to the intervalley scattering features. Note, in perfect graphene intravalley backscattering is in principle forbidden due to pseudospin conservation resulting in the absence of the central ring in FFTs, however its presence in G/Au flakes is discussed in section 8.1.
In Figure 7.10 (a) magnifications of the top left intervalley scattering circle are shown for bias voltages between−50 mV and+30 mV. The variation in diameter is clearly re-solved going from high bias voltages to lower bias voltages. In order to obtain the dis-persion relation, the radial distribution of intensity around the center of each interval-ley scattering ring was determined by azimuthal integration [Figure 7.10 (b)]. For the evaluation of the graphene dispersion relation in Figure 7.10 (c) all scattering discs were taken into consideration and subsequently averaged (given the contour is sufficiently well reproduced). The red squares show the corresponding dispersion relationE(k) of graphene for the voltage range between−50 meV and+50 meV. The graph shows thekG -vector of graphene, which is measured with respect to the K-point. The data clearly fol-lows a linear trend and a fit of the energyE(k)= ħvFk+EDyields a Dirac point shifted to ED=(+0.24±0.05) eV with respect toEFand gives an estimate ofvF=(1.1±0.2)·106m/s for the Fermi velocity. The observed Fermi velocity also nicely corresponds to the ex-pected value of 1·106m/s [7]. The parabolic dispersion relations of the Au surface state obtained at the edge of the same flake are plotted in cyan (Au) and blue (G/Au) including the Au surface state backscattering values extracted from atomically resolved maps. They clearly show the different behavior of the two electronic systems. Using the Fourier trans-form based determination of dispersion relations is very important for graphene, since only the correct assignement of the features in FFT allow for an unambiguous measure-ment of the graphene states. A simple measuremeasure-ment of the periodicity of standing waves in real space images is not sufficient, since one can see from the previous investigations that both intravalley backscattering features as well as surface state backscattering fea-tures yield similar real space modulations with different periodicities, only distinguish-able when considering the complete atomically resolved mappings.
The obtained dispersion relation shows ap-doping which is in agreement with re-cent ARPES studies [130, 251, 99]. Comparing the value for the Dirac point of 240 meV with the values of 100–150 meV [130, 251] obtained from ARPES studies for monolayer graphene on gold systems, shows a slight deviation. The ARPES experiments were per-formed on samples with large scale graphene on a single intercalated layer of gold, and thus show different system morphology, i.e. the absence of herringbone reconstruction,
7.2 Dispersion relations of single and double layer graphene 119
Au surface state and small flakes. However, various experiments performed on several other GNFs with varying sizes, various rotation angles between graphene and Au and also on significantly thinner intercalated Au layers show similar behavior and suggest the electronic properties and doping of graphene on Au to be largely independent of surface topography and surface states.
Concerning the shift in the Dirac point position, the electrostatic potential of the tip has to be considered since an influence cannot be ruled out. The small distance during atomically resolved mappings (the tunneling current ofI≈1 nA allows for an estimation of a tip to sample separation ofd=0.5 nm) and the very small DOS of graphene close to the Dirac point can affect the position of the Fermi level with respect to the Dirac point [252] when applying an electric field. Band bending due to tip electric field induced doping has been observed for graphene systems [253] and will lead to an increased slope of the dispersion relation, slightly overestimating the Fermi velocity. Furthermore, an additional doping can also occur only from the mere presence of an STM tip close to the surface without applying an electric field, because of the different work functions of tip and sample [73, 158]. Understanding of the influence of the tip work function on the Dirac point could be tackled by using different tip materials with large difference in work functions (PtIr, W).
7.2.2 Twisted bilayer graphene on Au(111)
Stacked graphene layers prepared by horizontal diffusion of intercalated flakes After the decoupling of GNFs from the Ir(111) substrate via intercalation of Au most GNFs are found in theR0° configuration preserving the registry of the Ir(111) substrate. The low rotational disorder of the GNFs on Au(111) was evidenced by LEED and discussed in chapter 6.1. However, in rare cases GNFs on the Au surface with orientations dif-ferent thanR0° are found, then often of thefloating type, thus all edges are on top of the supporting Au(111) surface. Occasionally, GNFs on the Au surface are encountered in stacked configurations commonly in combination with large rotation angles, hence suggesting that the annealing of the sample during intercalation involves a horizontal diffusion of GNFs in combination with rotations, if the flakes are not pinned.
In Figure 7.11 (a) the topographic STM image of two stacked GNFs is shown. A large floatingGNF marked with A covers the left side of a smaller GNF, which is embedded in the Au(111) surface marked by B. A magnification is shown in Figure 7.11 (b). The large flake A overlaps with theembeddedflake B as shown in the inset in the lower left corner and forms two stacked graphene layers. The two stacked GNFs are rotated with respect to each other, giving rise to an additional moiré superstructure on the area of the overlap-ping flakes marked with G/G/Au. Utilizing the atomic resolution on the G/Au region on the lower flake, the orientation was determined asR0° (the orientation was measured in the FFT via the angle between the superstructure reflexes of the herringbone
reconstruc-(b) G/Au G/G/Au G/Au
G/G/Au
10 nm
c d
e
(c)
(d)
(e)
50 nm
(a)
b A
B
R7 R0 2 nm
(f)
[11-20]
f
Figure 7.11 | Twisted bilayer graphene. (a)STM topography of a largefloating flake inR7.8◦ orientation overlapping with a smaller R0◦ flake. Inset: Schematic representation of the flake ar-rangement: Lower graphene layer red color, top most GNF, grey color. (b)STM topography of the overlap region.(c-e)Atomically resolved topographic magnifications: 4.3×4.3 nm2. The white arrow marks the carbon [1120] direction. (f)Magnification of the edge topography in the G/G/Au area with atomically resolved features. The edge is aligned along the zigzag direction. Scan parameters: (a) V=0.3 V,I=0.5 nA; (b-e)V=0.3 V,I=0.5 nA; (f)V=0.3 V,I=0.5 nA.
tion and the reciprocal lattice of graphene). The large flake on the other hand is tilted by an angle of∼7.8° with respect to the Au lattice. The rotation angle of the topmost flake suggests that the large flake has undergone horizontal diffusion in combination with a rotation during the post-annealing step of the intercalation based preparation procedure eventually ending up in a configuration partly covering the smallerR0° flake. The flake is pinned only by a small cluster just above the top of theembeddedGNF.
The magnifications in Figure 7.11 (c-e) show atomic resolution and moiré of the dif-ferent graphene areas: on theembedded R0° flake (c), on the rotatedfloating flake (e) and on the overlap region of both flakes (d). The white arrows mark the graphene [1120]
directions, which are clearly visible from the atomic resolution and illustrate the small twist in the orientation of the overlapping GNFs. Along with the small rotation angle of thefloatingflake, the moiré periodicity decreases compared to theR0° moiré (see equa-tion 2.22) leading to a subtle difference in appearance of the moiré in Figure 7.11 (c) and (e). More drastic changes to the appearance of the STM topography are found for the G/G/Au region, where the moiré becomes much more pronounced compared to the Shockley partial dislocation lines of the herringbone reconstruction. The moiré super-structure is now more irregular, presumably because it is a superposition of herringbone reconstruction, G/Au moiré of the first layer and G on G/Au, which also shows a moiré due to the rotation of the two graphene layers [254, 58, 239]. The twisted bilayer graphene moiré is also visible in the FFT of atomically resolved dI/dV mappings or topographies, as depicted in Figure 7.12 (c). The hexagonal superstructure spots are positioned around the center of the FFT and yield a moiré superstructure periodicity ofdm=2.02 nm.
Ac-7.2 Dispersion relations of single and double layer graphene 121
Figure 7.12 | Scattering in G/G/Au. (a)dI/dV mapping at +10 mV bias voltage across the G/Au and G/G/Au areas.(b)FFT of the G/Au dI/dV mapping.(c)FFT of the G/G/Au dI/dVmapping. Scan parameters: (a) 48.6×48.6 nm2,V=10 mV,I=1.0 nA,Vmod=3 mV,T=8.6 K; (b) Real space area of dI/dV mapping: 40×40 nm2; (c) Real space area of dI/dV mapping: 20×30 nm2.
cording to equation 2.23 this corresponds to a twist angle of∼7° similar to the twist angle determined from the orientation of the single flakes with respect to the Shockley partial dislocation lines.
Electronic structure of twisted bilayer graphene
It was experimentally found that achieving atomic resolution on the stacked GNF system is much easier compared to the G/Au system. However, the features observed regarding the scattering of edges and defects are similar for both the G/Au and G/G/Au system. A magnification of the right edge of the largefloating flake is depicted in Figure 7.11 (f ) in the overlapping area. Similar to the study of graphene edges on Au(111) and compa-rable to the DFT calculations of freestanding graphene layers in section 6.2.2, increased intensity in STM topographies at positions corresponding to the edge atoms of zigzag segments are found. Local intensity modulations of the atomic rows decay into the flake interior and show qualitatively similar features as comparable edge configurations for G/Au(111). This finding suggests a single H-terminated edge configuration as in the case of G/Au(111). Quantitatively the edges of the G/G/Au area show a larger enhancement of intensity compared to the edges in the G/Au area hinting at an even smaller interaction
0 0.1 0.2 0.3 0.4 0.5
Figure 7.13 | Dispersion relation of a twisted bilayer graphene on Au(111) from intervalley scattering. (a)Magnifications of the top right intervalley scattering disc for various energies (3.5× 3.5 nm-2). (b)Radial distribution of intensity after an azimuthal integration around the center of the ring in the FFT.(c)Dispersion relation of the twisted bilayer graphene (black) and of the monolayer graphene from Figure 7.10 (red).
of the in both cases quasi freestanding graphene edges.
Similar to the G/Au system, the characteristic intensity modulation of the edges is a clear fingerprint of the electronic structure of quasi-freestanding graphene. In order to investigate the electronic properties in more detail, dI/dVmaps were performed [Figure 7.12 (a)] and the FFT was investigated [Figure 7.12 (b,c)]. In the case of the G/Au area the features in the FFT are as in the previous investigations, showing a superposition of Au surface state scattering (A), intravalley (B), intervalley (C) and scattering around recip-rocal lattice points (D). The Au surface state backscattering is not visible in the G/G/Au area, hence indicating the decay of the surface state wave function in direction perpen-dicular to the surface to an amount small enough in order to be not observable anymore.
Both areas of the GNF, i.e. on G/G/Au and G/Au areas, show pronounced intervalley scattering reflexes (C) which were used to quantitatively determine the dispersion rela-tion of twisted bilayer graphene [Figure 7.13]. Atomic resolurela-tion in the dI/dV channel was achieved for voltages down to−140 mV. Magnifications of the intervalley features (C) at different bias voltages are depicted in Figure 7.13 (a) and the radius of the circu-lar features was determined using azimuthal integration around the center of the ring (b). The FFTs show only a filled disc in the voltage range above−20 mV due to the
lim-7.2 Dispersion relations of single and double layer graphene 123
itedq-resolution ofqmin=2π/L=0.3 nm−1withL=20 nm being the narrow side of the real space mapping area. Below−20 mV the intervalley scattering features become cir-cles, with a radius¯
¯¯~qinter−Γ# –K¯¯
¯=2kwhich increases for growing energies in the occupied states. The dispersion relation is plotted in Figure 7.13 (c), black curve, including only those voltages, where a clear ring could be resolved. The linear fit yields a Dirac energy ofED=85±25 meV and a Fermi velocity ofvF=(0.8±0.1)·106m/s. For comparison the dispersion relation of G/Au(111) from the previous chapter is also plotted in red. A severe shift in energy is immediately visible, when comparing both dispersion relations, result-ing mainly from the downshift of the Dirac point closer to the Fermi level (G/Au: 240 meV compared to 85 meV for G/G/Au). This downshift is readily understood by a better de-coupling of the second graphene layer in conjunction with an electrostatic screening of the doped first graphene layer comparable to graphene multilayers on the carbon face of
¯=2kwhich increases for growing energies in the occupied states. The dispersion relation is plotted in Figure 7.13 (c), black curve, including only those voltages, where a clear ring could be resolved. The linear fit yields a Dirac energy ofED=85±25 meV and a Fermi velocity ofvF=(0.8±0.1)·106m/s. For comparison the dispersion relation of G/Au(111) from the previous chapter is also plotted in red. A severe shift in energy is immediately visible, when comparing both dispersion relations, result-ing mainly from the downshift of the Dirac point closer to the Fermi level (G/Au: 240 meV compared to 85 meV for G/G/Au). This downshift is readily understood by a better de-coupling of the second graphene layer in conjunction with an electrostatic screening of the doped first graphene layer comparable to graphene multilayers on the carbon face of